Permutations with equal orders

Abstract

AbstractLet${\mathbb{P}}(ord\pi = ord\pi ')$be the probability that two independent, uniformly random permutations of [n] have the same order. Answering a question of Thibault Godin, we prove that${\mathbb{P}}(ord\pi = ord\pi ') = {n^{ - 2 + o(1)}}$and that${\mathbb{P}}(ord\pi = ord\pi ') \ge {1 \over 2}{n^{ - 2}}lg*n$for infinitely manyn. (Herelg*nis the height of the tallest tower of twos that is less than or equal ton.)